The Brunn–Minkowski Inequality and Nontrivial Cycles in the Discrete Torus

Let $(C_m^d)_{\infty}$ denote the graph whose set of vertices is $Z_m^d$ in which two distinct vertices are adjacent iff in each coordinate either they are equal or they differ, modulo $m$, by at most 1. Bollobás, Kindler, Leader, and O'Donnell proved that the minimum possible cardinality of a set of vertices of $(C_m^d)_{\infty}$ whose deletion destroys all topologically nontrivial cycles is $m^d-(m-1)^d$. We present a short proof of this result, using the Brunn-Minkowski inequality, and also show that the bound can be achieved only by selecting a value $x_i$ in each coordinate $i$, $1\leq i\leq d$, and by keeping only the vertices whose $i$th coordinate is not $x_i$ for all $i$.